MA572 MAPLE Example Sheet 1 If this is the first time you have ever

 !#"%$'&)(+*,(-*,$.&)/1032.*4$5$'(-6/798;:<&)=?>9/@/A>;/B2C:D*./EFG=9H)I+/9J3KMLONPLPQSRUTPNC!4VW+WYX9:)*4$5$Z7[H\/M(-]^$'&D/_*.=96/
` 8;6a6a=9])ED*b=9*=c2./d(+]e$'&)(+**f&)//A$gJh$'86a=9i9/j*.:)2f/j7;8;:lk;/A$$.&)/d*'=c6a/^=c])*fmn/2f*opQ5&)/]l=9E)=9HD$b$.&)/
k;(q>;/B] ` 8rE)/a$.8*.8;Iq>;/j$.&)/jHD2.8;s)I+/6*b=c$@$'&)/a/])EtoQ%8G*.//=FG=9H)I+/duv/BI-Hp03I-/j=9sU89:D$bH)I+89$'*jwx*.=g7)yzJ
*f(-6H)I+7a$Z7[H\/M(-]{}|~r€‚ƒ
„†… v ‡‰ˆ Še1‹Œ
VY8;6H)I+/AŽ]:)6sU/B2.*(-]a‘’U“x”n=92f/5E)/B03]D/Ej:D*.(-]DkP$'&)/•*f7r6bs\8;I—–J;m•&)( ` &a(+*˜/™:)(q>c=9I-/B]$h$.8ƒ9šœ›œ.ž\Ÿ; o
VY8;6H)I+/AŽ=92f(+$.&)6a/A$'( ` (-*5H\/24"¡8;2.6/E=9:[$'8;6a=c$'( ` =9I+I+7¢8;] ` 896aH)I+/AŽ^=92.k;:D6a/B]$.*o¤£38;2,/zŽ)=c6aH)I+/
¥ ¦n§Z¨ .©œª«)¬[– œ¬­4®[ªr¯D¬r– ±°
²¢³µ´·¶P¸•¹»º;ºc¼
¥ Ÿc½ ¦ °
¶ ¾9¿;¸ ¿ ¶ ¾9º9¿9º ¿ ¼
À (+>9/]= ` 8;6aHDI-/AŽa]:)6bs\/2BJD/coÁk)o ²´ ¿ ¹p (ÃJrQ5&)/M‘B’t“x” ` 8;66j=c])E)*nÄDÅ¢=9])E–Æj03])Ed$.&)/P2./=9IU=9])E
(+6j=9k9(-]3=9247H3=92f$.*•89"˜= ` 896aH)I+/AŽd]r:D6bsU/B2J32f/*.H\/ ` $'(q>;/Iq7;o
¥ ¦n§Z¨ f«rªœ®3¬r– ±°
²a³µ´ ¿ ¹Â9¼
¥ ÄDÅǝ ¦ ±°
¿
¥ –Æn ¦ ±°
Â
Q5&)/b‘’t“¡” ` 8;66j=c])E)*vÈcÉUƒa=9]DEÈ;›Ê;ˏÆ3ÅcÌr<0)])E<$.&)/@68rE):)I-:D*wx=9s)*f8;I-:D$./1>9=cI-:)/gy•=c])E=92fk;:)6/]œ$gJ
2f/*.H\/ ` $'(q>;/Iq7;o
¥ ÈcÉ\ƒ­ ¦ ­°
¾
¥ È;›Ê;˜ÆÅcÌr ¦ ­°
=92 ` $Í=9]ÏΠ¿UÐ
Q5&)/‘’U“x” ` 896a6a=9])Eќ€}Ì3ÒcË[Ê[È9)Åj$'=9i9/B*•$.&)/ ` 8;6H)I+/AŽ ` 8;]‚Óf:)kœ=‚$'/189"Ç= ` 8;6H)I-/zŽ^]:)6bs\/2Bo
¥ ќ€}Ì)ÒcËDÊ[È9DÅǝ ¦ ­°
¿ ¶ÔÂ;¼
Q5&)/‘’U“x” ` 896a6a=9])Eќ€}Ì[ÕDÅ;›œH):D$.*v= ` 896aH)I+/AŽd]r:D6bsU/B2•(-]œ$'8¢H\8;IÖ=92n"¡8;2.6 ³
¥ ќ€}ÌrÕDŜ›œ ¦× |€~;È;›\ ­°
H\8;I-=92 Î ¾[Ø =92 ` $Í=c] Π¿ бÐ
Q5&)/‘’U“x” ` 896a6a=9])EÅ9ÕDÈ[~[Ñb/B>c=9I-:)=c$'/B*•=9]</zŽ[H)2./B*.*.(+8;]:)*.(+])k ` 8;6H)I-/zŽ=92f(+$.&)6a/A$'( ` =9])E ` 8;]œ>;/24$'*
(q$5$'8 ` =924$'/B*.(Ö=c]"¡8;2f6 ³
¥ Å9ÕDÈ[~rѱ4Ù ­°
¿ ¹Â;¼
Ú
Û oÜr:)H)H\8;*./ ²b´ ÞfßDÝ àâá J)m•&)/B2./@ãd=9])Eä1=92f/12./g=cIå]r:D6bsU/B2.*BoY£Ç(+])E^$'&)/M2f/g=9I­=9]DE<(+6j=9k9(-]3=9247H3=92f$.*
8c" ² on£Ç(+])Ed$'&)/M68[ED:)I-:)*•=c])E=92.k;:D6a/B]$C89" ² yzo
¥ ¦n§Z¨ Ÿc½%.Ȝª3–œ¬}ÉU ­°
Ú
²¢³æ´ ã ¹¼ ä
¥ Å9ÕDÈ[~rѱ ¦ ­°
ã
¼ä
¶
ãç ¹ äÍç ãœç ¹ äÍç
¥ ÈcÉ\ƒ­ ¦ ­°
Ú
è ã ¹¼ ä è
¥ Å9ÕDÈ[~rѱ4Ù ­°
Ú
é ãœç ¹ äÍç
¥ È;›Ê;˜ÆÅcÌr ¦ ­°
=92.k9:)6a/B]œ$•Î ã Ú ä Ð
¹¼
9
Å
D
Õ
[
È
r
~
±
Ñ
4


Ù
­
°
¥
=92 ` $'=9]­w ¶ ä Ø ãry
ê Nv89$./<$.&3=c$j"¡892a2f/g=9Iv=92.k9:)6a/B]œ$'*aë­Jíì\J,$'&)/$ZmY8Ô=c2.k;:)6/]œ$a"¡:)] ` $.(-8;]î=92 ` $Í=9]åwxì Ø ëUy ` 8;6H):D$'/B*a$.&)/
H)2f(-] ` (-H3=cIt>c=9I-:)/18c"%$.&)/_=92.k;:D6a/B]$•8c"%$.&)/ ` 8;6H)I-/zŽ^]:)6bs\/25ë ¹ (-ìa*.8 ¶Cïñð =c2 ` $'=9]­w¡ì Ø ëUyYò ï oíó
ôöõ)÷%øxùpúû\üýüñõ3þ%ÿ ôöùõ3þ þ ÄDÅǝ ¦ £Ç(+])E)*5$'&D/_2f/g=9ItH3=92f$C89"Ç= ` 8;6H)I+/AŽ^]:)6sU/B2 ²
–Æn ¦ £Ç(+])E)*5$'&D/_(+6j=ck;(-]3=c2f7¢H3=924$C89"˜= ` 8;6aHDI-/AŽd]:)6bs\/2 ²
ÈcÉ\ƒ­ ¦ £Ç(+])E)*5$'&D/@=cs)*.8;I+:D$'/P>c=9I+:)/189"Ç= ` 8;6H)I-/zŽ^]:)6bs\/2 ²
È;›Ê;ˏÆ3ÅcÌr ¦ £Ç(+])E)*5$'&D/@=c2.k;:)6/]œ$•89"Ç= ` 8;6H)I+/AŽ^]:)6sU/B2 ²
ќ€}Ì)Ò9Ë[Ê[È9D؝ ¦ £Ç(+])E)*5$'&D/ ` 8;]‚Óf:)k;=c$'/189"Ç= ` 8;6H)I-/zŽ^]:)6bs\/2 ²
ќ€}ÌrÕ)Å;›œ ¦¤× |€~;Ȝ› VY8;]œ>9/2f$.*P= ` 896aH)I+/AŽd]r:D6bsU/B2 ² (-]œ$.8H\8;I-=92Y"¡8;2f6
Å9ÕDÈ[~[Ñ­ ¦ VY8;]œ>9/2f$.*P= ` 896aH)I+/AŽd]r:D6bsU/B2 ² (-]œ$.8 ` =92f$./*f(Ö=9]^"¡8;2f6
… ˜ •… v … % F =cH)I-/Á*P|~€c ` 8;66j=9]DEp(+*P"¡8;21H)I+89$.$.(-])k^2f/g=9I˜"¡:)] ` $'(+8;])*189"Y=2./g=cI˜>9=c2.(Ö=cs)I-/co£38;2 ` 8;6H)I+/AŽHDI-89$.*
G
$.&)(-*5(+*,6a89*f$'Iq7j(+]3=9E)/B™:3=c$'/co!Z]$.&)(-*5*f/ ` $'(-89]!¤k;(q>;/1*.896a/P/zŽD=96aHDI-/*58c"Ç*f8;6a/ ` 8;66j=9]DE)*5=9])E&)8?m
$.&)/B7<=92f/@:)*f/E—oCF8;2./_(-]D"¡892.6a=c$'(+8;]=9s\8;:D$C&)8?m $'8a*.H\/ ` (+" 72'=c])k;/*BJ ` 8;I+8;:)2.(+])k)J3=c])EG*f8j8;]—J ` =9]Gs\/
"¡8;:D])E<(+]^$'&)/M8;] #I-(+])/P&)/BI-HH3=9k;/B*o
!#"%$
V,=9*f*.(+])(Ö=9] ` :D2f>;/B*=c2./M*.8;I+:D$'(+8;])*,8c"
è ²M¶ ã è'&)è ²M¶ ä è ´ ` 8;])*f$'=9]œ$ ´)( ç Ø
$.&3=c$_(-*BJ—$'&)8;*f/ ² "¡8;2Pm•&D( ` & $'&)/¢H)2f8[E): ` $_89"¤$'&)/¢E)(+*f$Í=c] ` /B*M"¡2f8;6 ² $'8ã=9]DE ² $'8ä(+* ` 8;]D*f$Í=c]$J
m•&)/B2./ ² J\ã3J3!ä *+houv/B2./1mn/@(+]>9/*4$'(-k;=c$'/1$'&D/ ` :)2f>9/*•m•&)/]Gã ´ Ú Jä ´ ¶ Ú o
¥ |—Ÿ §Z¨ Å;ÕDÈ[~rÑ­'ÈcÉ\ƒ­ ,rª3–œ.¬ -\0 /‚©žUŸ;  ­°
132³µ5
´ 4 7ë 6 ¹»º ë ç ì ç ¶º ë ç ¹ 0ì 6 ¹»º ì ç ¹ Ú
º
5Q &)(+*@k9(+>;/B*@:D*M$.&)/™r:)=9]œ$'(+$Z7 è ²b¶ Ú è+è ²M¹ Ú è ´ è ² ç ¶ Ú è "¡8;2 ²<´ ë ¹ (+ìUoNC/AŽr$gJ%(-] 8;2.ED/2M$'8H)I-8c$
"¡8;25m•&)( ` &ë­Jìd(+]^$'&)/aw ` 8;6H)I-/zŽ3y,H)I-=9])/P$.&)/M™r:)=9]œ$'(+$Z7d|—Ÿ_$'=9i9/B*C>c=92f(-8;:D* ` 8;])*4$Í=9]œ$•>c=9I-:)/B*J)mn/@:D*./
/ 9.;=<?>A@B;=CD;=@FEG;=HC(-*v$.8ds\/bH)I-8c$.$'/BE—J\*.8
8 Ær|~ 8 Ñ 8 œ|\~€‚\oQ5&)/@032.*4$M=92.k9:)6a/B]œ$P(-*C$'&)/™r:)=9]œ$'(+$Z7m•&)8;*f:
(+]d$'&)(+*5])/AŽr$•H)I+89$,mn/@=c2./P*f//i(+])k$.&)/1I-8 ` :D*,89"±HU89(-]œ$'*,*f: ` &<$.&3=c$Y|—Ÿ}žŸ ¨ I o¤Q5&)/1*./ ` 8;]DEG=9]DE$.&)(-2fE
=92fk;:)6/]œ$'*=92./M$'&D/2'=c])k;/*C"¡8;2•ë =9])Eì^2f/*fHU/ ` $'(q>;/BI+7;oCQ5&)/@Êr› 8'J 8;HD$'(+8;](-*Cs7E)/A"x=9:)Iq
$ K ©¯ × ©rM¯ LON
!n&3=g>;/_(-] ` 2./g=c*./E$'&D/_]:)6sU/B2C8c"ÇH\8;(-]œ$.*,$'/*4$'/BE<sœ7<FG=9HDI-/P$'8¢k;/A$v=¢s\/B$f$'/B2C2f/*f8;I-:D$.(-8;]pw¡I-/=?>9/1$'&)(+*
8;H[$'(-89]89:D$C=9])E7;8;3: ÁI+Iå*.//1m•&3=‚$v!n6a/=9]yzo
£Ç(+2.*4$gJ•$'&)8;:Dk;&—Jvmn/ &3=g>;/ $'8Ô$'/BI-IPFG=9H)I+/$'&3=c$^mn/m•(-I+Is\/p:D*.(-]Dký$'&)/ 68;2./=cED>c=9] ` /BEHDI-89$f$'(-]Dk
H)2f8 ` /E):D2./*BJUsœ7^I+8œ=9E)(+])k$.&)/P|\~€‚ƒ¢H3= ` i‚=9k;/co Pe/ME)8¢$'&)(+*5sœ7^$Z7[HD(-])Rk Q 8 T S¤Z|~€cƒr §
¥ Q 8 U S¤ |~€‚ƒ[ §
¥ 8 Ær|~ 8 Ñ 8 œ|~r€‚Z|—Ÿ‚žŸ × , ¨ ž;%® VTV ® × - ¨ ž;%® VTV ® × Êr› 8'J ¨ KAŸ ITIh× Ÿ ITI L\ ­°
0.4
y
0.2
–1.4 –1.2
–1
–0.8 –0.6 –0.4 –0.2
0.2
0.4
0.6
0.8
x
1
1.2
1.4
–0.2
–0.4
W¤8;: ` =9]a*.//v$'&3=c$n])/=92¤$.&)/892.(-k9(-]—J;$.&)/H)I+89$¤(-*¤(-] ` 8;2.2f/ ` $go˜!Z]j"x= ` $gJUwYX Ø Xœyh*.&)89:)I-EasU/v8;]a$.&)/vk92'=9H)&to
Zn7ý*.I-(+k;&œ$'Iq7e=cE‚Óf:)*f$.(-])k$.&)/^k;2f(-EÔ]:)6bs\/2f*JhmY/ ` =9]=9I+$./2m•&)( ` &lHU8;(+]œ$'*¢FG=cH)I-/^(+*$'/B*f$'(+])k)opQ5&)/
2f/*.:DI+$C*.H\/g=9i*•"¡8;25(+$.*./BI+" o
!¤$'&)/B]<H)I+89$5$'&D/_I+/B>9/Iå*./A$C89"\[ Ý ¶ MÚ ]µ¾ ´ XDo
¥ 8 Ær|~ 8 Ñ 8 œ|~r€‚Z|—Ÿ‚žŸ × , ¨ ž;%® VTV ® × - ¨ ž;®%VTV ® × Êr› 8'J ¨ K_^¯ × ^r¯`L ±°
¿
0.4
y
0.2
–1.4 –1.2
–1
–0.8 –0.6 –0.4 –0.2 0
0.2
0.4
0.6
0.8
x
1
1.2
1.4
–0.2
–0.4
¥
8
Ær|~ 8 Ñ 8 œ |~r€‚Z|—Ÿ‚žŸV ¯ × , ¨ ž;®aVTV#® × - ¨ ž;®aVTV#® × Ê› 8.J ¨ K ¯ Ih× ¯ I L ­°
0.6
y 0.4
0.2
–1.6 –1.4 –1.2
–1 –0.8 –0.6 –0.4 –0.2 0
0.2
0.4
0.6
0.8
x
1
1.2
1.4
1.6
–0.2
–0.4
–0.6
Q±8a*f//_6a892./P$'&)=9]8;])/_=c$C=$'(-6/9JrmY/ ` =c]:D*./@= ` 8;]œ$'8;:D2CHDI-89$o
¥ {rќ€}Ìr€}Ë[›;|~r€‚
Q5&)/P"¡89I-I-8?m•(+])k ` 8;66j=c])EjH)I+89$'*¤$'&)/ ` 8;]œ$.8;:)2.*n"¡8;2¤|—Ÿ ´ Ú'b º J Ú J ¿Ub º J º J ¿ =9])E  m•(+$.&)(-]$'&)/P2.=9])k;/B*
"¡8;2,ë=9])Eìdk;(q>;/]tJ)m•(+$'&/B])8;:)k;&2./B*.8;I+:D$'(+8;]d$'8¢*f//M$'&)(+])k;*5H)2f8;HU/B2.Iq7;o
Â
¥ ќ€}Ìr3€‚Ë[›;|~€cZ|—Ÿ × , ¨ ;ž ®%V0V#® × - ¨ ;ž ®%VV#® ×
я€}Ìr3€}ËD›3ƒ ¨ KzŸ9½;© × Ÿ × D« ½;© × © × « × ® L × Êr› 8'J ¨ KZ®[¯ × ®D¯ML ±°
1.8
1.6
1.4
1.2
1
y
0.8
0.6
0.4
0.2
–2
–1
1
–0.2
x
2
–0.4
–0.6
–0.8
–1
–1.2
–1.4
–1.6
Ï1‹ gŒÇ1Œ
Û odcC*./p‘B’t“x”d&)/I+H $'8ýk;/A$^68;2./<(-]D"¡8;2f6j=‚$'(-89]î=9s\8;:D$a$'&)/‘’t“¡” ` 896a6a=9])E)*ÄDÅ3J_–ÆtJPÈcÉ\ƒrJ
È;›Ê;˜ÆÅcÌr\J±Ñœ€}Ì)ÒcËDÊ[È9DÅ)J—È;›3Ñ})ÈcÌUJåќ€‚ÌrÕDÅ;›œ½}|€~;Ȝ›J—Å;ÕDÈ[~rÑro
o cC*./‘’t“¡”•$'8¢/B>c=9I+:3=c$./
e d
wYfZy Î ÚÚ ¹ (( Ðhg Ø
¶
wYfif#y
j
Ú ¹ ( é ¿ ÝYm
é ¿ ¶ l( k
o cC*./‘’t“¡”•$'803])E$'&D/_2f/g=9I­=c])E<(+6j=ck;(-]3=c2f7H3=924$'*•89" ³
n d
é
é
wx(Öy w Ú ¹ ¿ (Öyw P ¶ ¾ ( y Ø wx(+( y ¿ Ú ( Ø wx(-(+( y ¾ 1¸ ¶ ¿ ( ( ¶ ¿º5¹Â¾ (( Ø wx(+>Dy w ¿ ¹ é ( yB¿ w º¶ ( ( ¿ y
¶ Â
ý
¹
¶
¶
]
o cC*./‘’t“¡”•$'803])E$'&D/_68rE):)I-:D*•=9])E<=92.k;:D6a/B]$•8c" ³
o d
{
w¡( y Ú ¹ ( é D¿ Ø wx(+( y w ¿ Ú ( y ç Ø wx(-(+( y Ú Ú ¶ é ¾ ¹sr Ú X ¹ (ut º é ¾.vwvØ w¡(+>DyÏ/zŽDHxt yÝ ï ( vz]
¶eÂ
Âqp
oÜr:)H)H\8;*./}|˜w¡ëUy ´ Ú w 8;*)ë Ú (?*.(-]nëUy4ç ocv*f/@‘’t“¡”v$.803])E$'&)/M2./=9I­=9])E(+6j=9k9(-]3=9247H3=92f$.*
¹ ` ¹
8c" |˜w 6Ý ï y5=9])~
E |˜w€6 ï yAo
¾
o cC*./»‘’U“x”p&)/BI-Hö$'8
 d
30 ])E 8;:D$=9sU89:D$|~r€‚ƒrJ 8 Æ[|~ 8 Ñ 8 œ|~€‚UJќ€‚Ìr3€}Ë[›œ|~€‚\o Q±8 03])E
89:D$_=9s\8;:D$1>c=92.(+8;:)*1H)I+89$.$.(-])kj8;HD$'(+8;])*M=9]DE ` 8;I-89:)2.(+])kd6/ ` &3=9]D(-*.6*P$Z7rHU/^{}|~r€‚‚Kf€}|[ 8 €}Ì\ƒLJ
{}|~€‚aK.ќ€~€9T› L
o cC*./‘’t“¡”•$'8¢H)I+89$5$'&)/MI+8
ƒ d
w =;y
w¡sy
w` y
` (t89"
è ²_¶ Ú ¶ ( è ¹ è ²¹ Ú ¹ ( è ´…„ "¡8;2 „j´ ¾[Ø Â Ø'¿ o
P &)=c$C(-*,$'&D/_6(-]D(-6b:)6
>c=9I+:)/189" „ "¡8;2,m•&)( ` &$.&)/MI-8 ` (—(+*5])8;]#/6HD$Z7†
è ²_¶ Ú ¹ ( è ¶ è ²¹ Ú ¶ ( è )
´ „ "¡8;2,>c=92.(+8;:)* „ o
è ²_¶ ( è-è ²•¹ Ú è ´ Ú
‰
KP2o#‡I+(Bˆ=9sU/A$'&<FG=9]D*f03/BI-E
º9º L ` $'8;s\/2 º XDX Ú